 Welcome back to this NPTEL course on game. In the previous sessions we have introduced certain learning methods. In this session, we introduced an important class of games called potential games. So, these potential games are very important because of their applicability in variety of pictures. So, in this session we discuss about this potential games and give some examples. The more details of this, we will not do it and instead we ask people to look at the literature. So, let us start. First, let us start to give the motivation and other things. Let us start with a game, a finite game with strategy, let us take a finite game with strategy sets, sets xi. For if we are taking only two players x1 and x2 are the two strategy spaces, if we are taking a multiplayer game that means x1, x2, xn provided it is an employer game. So let us assume all players have the same payoff that is there is a b from x to r such that ui x1, x2, xn. Let us assume n player game. Player one strategy is x1, x2 is player 2, xn is n players. This is simply given by px1, x2, xn. That means every player has the same payoff function. Now, how do we find an ashtrayam of this? So, if we really look at it little carefully, so look at take x bar which is x1, x2, xn which is inside the arg max of px1, x2, xn where x belongs to capital X. Look at the one x which maximizes function p and then take that x bar. Basically x bar maximizes this. Then you can easily verify that then x bar is Nash equilibrium in pure strategies. So, this is an instance where you can show that the game admits a pure Nash equilibrium and that pure Nash equilibrium can be computed simply by a maximization problem. This is an interesting example. In fact, one thing would like to point out is that there may be other Nash equilibrium in this game. This x bar is not the only Nash equilibrium but there will be other Nash equilibrium. They may give less payoff compared to this one. This is the best that they can get. Now, let me look at what is called best response dynamics. This is another class of learning dynamics. Let us look at this one. So, start from an arbitrary strategy profile x1, x2, xn. Then next thing is ask if a player i has a strategy xi prime which gives better payoff or basically strictly increases his payoff. That means ui xi comma x minus i is strictly greater than ui xi x minus i. If s replace xi with xi prime, otherwise we are at Nash equilibrium. So, if there is no strategy which gives him better no player has this thing then that is automatically Nash equilibrium. So, if there is a strategy which gives higher payoff then just follow. So, what is really happening is that this particular thing in this best response dynamics what you are doing is that every time you are trying to increase. Now, if we have this the game like in previous setup if the game have the same payoff function what is really happening here is that each time there may be an increase for anyone the number of improvements like improvements will be finite. So, eventually it will lead to a strategy where no one can improve it. So, therefore we will arrive at an equilibrium. Remember this we are arriving at an equilibrium here mainly because the set of strategies available is finite and then the game is admitting a pure equilibrium pure Nash equilibrium here. So, and the number of pure strategies is finite. So, whatever wherever you start with it and if you improve it and at any point of time the number of choices available is finite and then overall this algorithm this procedure will run at most finite times and even before it reaches to the Nash equilibrium. Let us look at payoff equivalence. Of course, the best response dynamics for general games is again a non trivial problem that need not converge. There are whatever I have said is a discrete version it can actually have a continuous analog as like in BNN dynamics that we have seen earlier. So, we can do it but those are all not certainly part of this course they are still an active area of research. So, now let us consider a general general game with payoffs ui from x to r we are assuming finite game that means then finite stress space. So, let us consider suppose if we add a constant to a payoff. So, that means let us consider ui bar x1 x2 xn is nothing but ui of x1 x2 xn plus some constant Ci. So, what is going to happen? Now what if what if Ci is basically a Ci of x minus i. So, there are two questions here one is if the ui bar is simply a the payoff that you are getting in this game ui plus some constant which depends on i. And the second question is what if if Ci is a function of x minus i. Now in fact you can easily see that when you do this one the best response is simply the same as the original game ui. Whatever maximizes ui when the x minus i is the other players are fixed strategies are fixed the same thing will maximize ui bar also whether Ci is a constant or Ci depends on x minus i. Therefore, the best response says are the same thing. So, therefore in fact the game Nash equilibrium structure remains the same thing whatever is Nash equilibrium for ui will also be Nash equilibrium for ui bar and this thing. In fact we will say that the payoffs ui bar and ui are said to be difference equivalent for player i the difference ui bar x1 x2 xn minus ui x1 x2 xn this is just simply a function of Ci x minus i does not depend on her decision xi but only on the strategies of other players. You say that the two games are difference equivalent if this ui bar minus ui is a function of simply x minus i and in a sense the payoffs the difference does not depend on the decision xi it depends only on the other decision. So, as I said as I pointed out already when you have this the Nash equilibrium structure remains unchanged. We can prove the following theorem find it games with div equivalent payoffs have the same u Nash equilibrium. The proof is as I said it is trivial and we can it is not hard to look at it. So, just I would like to introduce the notation. So, delta f xi prime xi x minus i I will put it as f xi prime x minus i minus f xi x minus i. So, the diff equivalent means actually what we are saying is that delta u bar of xi prime xi x minus i is same as delta u delta ui xi prime xi x minus i. So, this is the diff equivalent using this notation introduced here delta. Now we are ready to introduce potential games. A game is called potential if it is diff equivalent to a game with common payoffs. So, this is diff equivalent with a game with common payoffs what it means is the following thing that is there exists a function p from x to r such that for each i x minus i in x minus i xi prime xi in xi we have delta ui xi prime xi x minus i is same as delta p xi prime xi x minus i when the the difference between the payoffs xi prime x minus i minus let me write it here this is nothing but ui xi prime x minus i minus ui xi x minus i this is same as p xi prime x minus i minus p xi x minus i if this condition holds then you say that this game is a potential and then the p is the potential function. Now in fact from whatever arguments that we have seen already but it says that the if a game has a potential function it admits a pure Nash equilibrium at least there is at least one pure Nash equilibrium and the best response iterates converge. So, the potential game is one place where the best response iterations converge. In fact I would like you to connect this with the fictitious play which I will leave it to you for thinking. So, let us look at some examples now. So, take the the game so consider this game 10 10 0 11 11 0 1 1. So, this is a bi-matrix game in fact verify that 0 1 1 2 adds as a potential for player 2 the differences when the first row is fixed let us say player 1's first row is fixed and then the difference of the player 2 is that 11 minus 10 which is same as 1 minus 0 and then when the second row is fixed this is by player 1 then what we will get is this is nothing but 1 minus 0 that is nothing but 2 minus 1 here and similarly for player 1 verified. So, this gives you an example of a potential game where this thing there is another very very important class of games which are potential game is basically a routing game. So, let me give this details. So, let us assume consider there are n drivers traveling different origins and destinations in a city. So, in a city there are several paths and then let us assume there are n drivers who are traveling from their origin to their respective destinations. So, they may be different. The transport that corresponding transport network is given by a graph n, a, n is basically the set of all nodes and a is the arc basically let us say that means suppose this is one node when I say this is there that means from this place to this place there is a path and then this is there let me take this. Now, for example, from if one player's origin is this and the destination is this he has to go only through this way and of course it is possible that if it is there then he can also use this one. So, for example if the origin is this place destination is this then this particular driver has two paths one is going along this and other is going along this there are two paths for him. Now, because of this congestion, congestion is a very common issue because of multiple player people are using the same road then congestion happens that travel time of an arc is non-negative increasing function. So, ta is basically ta of na of the load na which is number of drivers using arc a in that specific arc a is used by the na drivers then the travel time it depends on this na which is a non-negative increasing function as na increases ta na increases. For example, one pure strategy j for i is a route a1, a2, al that is a sequence of arcs connecting his origin and the destination that the travel time is going to be which is the ui r1, r2, rn is going to be summation a in ri ta na. So, basically na is nothing but number of j in a in rj. So, this is the way you will define this their pay off functions in fact they want to minimize the travel. So, therefore, these are the cost minimization problems and not pay off this is the cost here they want to minimize the travel time. So, this is there. So, this is basically known as the routing game in fact one very interesting thing is that there is a theorem which is due to Rosenthal is the following thing. Routing game has a potential given by e r1, r2, rn that is nothing but summation a in a summation k is equals to 0 to na ta k where of course na is nothing but number of j such that a is in rj. So, this routing game is an example of a potential game. This is a very important class of games and there is a lot of research happening around this particular class of games. So, the proof is actually a trivial thing which is not really difficult to see this one. Then there is another game I will introduce and then is congestion games. In fact, congestion game includes the routing game this is a more general than the routing game in fact routing game is a special case of congestion games. So, there are each player i1 to n has to perform a certain task which requires some resources taken from a set r. So, this set r contains certain resources and each player is to perform a task which requires some resources from r. The strategy set xi is basically a for player i contains all subsets xi which is contained in r because his strategy is basically a subsets of r. So, that is this thing. Now, each resource has a cost crnr where nr depends on the number of players using the resource. If the nr is basically the number of people who are using the resource r. So, the cost depends on the number of people who are using that resource. So, players pay only for the resources that he or she uses. Therefore, ui x1 x2 xn is nothing but r in xi crnr of course nr is nothing but number of j such that r is in xj. Now, this is known as a congestion game. In fact, we can say that this is a potential game. So, which I will leave it as an exercise for you. So, we can verify that this is a potential game. In fact, there are a lot of work that is happening around these potential games. And one of the very interesting with this potential game is that the fictitious play for example converges. So, that is one very advantageous, advantageous property of this potential games. And they also enjoy a lot of other interesting theoretical properties. One of the very simplest thing is the best response dynamics is converging that we have seen it already and then existence of pure equilibrium and things like that. And it has found several applications. So, which we have seen couple of examples from traffic networks. So, which is a in fact, there is a lot more to this subject than what I have presented here. Okay, with this I will stop this and we will continue in the next session with cooperative games.